1549301742-The_Theory_of_Difference_Schemes__Samarskii

22 Preliminaries

can be made self-adjoint:

Ayi = aitl (Yitl - Yi) - ai (Yi - Yi-i) - di Yi= -<pi,

i = 1, 2, ... , N - 1.

Indeed, we multiply equation (36) by the function T/i f. 0 and involve

the conditions Ai T/i = ai and Bi T/i = ai+I, whose use permits us to establish

Ai+ 1 T/it 1 = Bi T/i = ai+ 1 and

Bi

T/itl = ~A 17i = T/1

it 1

where 7) 1 is an arbitrary constant. As a result we get the equation A Yi =

-<pi with <p; = TJi Fi and a;= Ai T)i, di= (Ci - A; - B;) 17; =Ci - ai - a;+1,

Ci = Ci T/i.

By applying the elimination formulae to the proble1n

(37) Ayi = ai Yi-1 - Ci Yi+ aitl Yitl =-<pi,

i = 1, 2, ... , N - 1, Yo= 0,

we deduce the estimate

(38)

under the natural premises

(39)

The computational formulae of the right elimination method help derive
estimate (38):

Yi = CYitl Yitl + /3itl ,

(40) O'.itl

/3i+l = ----ai /3i + <pi

Ci - O'.i ai

i = 0, 1, ... , N - 1,

i=l,2, ... ,N-1,

i=l,2, ... ,N-1,